Relay Driver Idea

Alright so it has been years since I have gotten on here and I'm not in the industry anymore so I don't care about IP so...

After goofing around my current home project which included driving a relay I had noticed that having built a custom assembly (pull-type solenoid w/ plunger that normally floats in and out, compression springs, and a custom made bracket to hold it all together) that the initial magnetic flux leakage was too great to overcome the spring constant with little to no alternatives...

So I tried rigging up a setup that uses the only available power and gives it a boost during initial turn on to overcome the small magnetic force at the initial large stroke distance. This was interesting because although I am a software guy as well, this is one of those situations where the uC alone can't save you (you need more than what the supply can give you so Duty cycle means squat).

Like any circuit, there are draw backs like the off time required to charge the charge pump back up (in this case roughly 20ms) but if you have an On Time like most setups this isn't really a problem factoring in Duty cycle. I thought it was neat because you can size the charge pump to droop after an initial time of charging to prevent the solenoid from overheating and give it a big boost at turn on without having to do anything extra in software or worry (it's always a safe failure condition with the charge pump in regards in feeding it a supply voltage).

Thoughts?

[EDIT] ignore the last npn gain because I meant for it to be a Darlington but simulated with something with much less gain (which is why it's completely saturated)

I keep running into the fact that a combination of storage charge or a higher charge pump voltage is required to outweigh the load inductance initial transient impedance.... annoying. Otherwise it only speeds up the switch but doesn't add any ooooomph

So I took a look at the fuel injector IC and it looks pretty cool but it can't change the poles in the system to do "more" than what bus allows. In fact it does the opposite, it was made to limit the holding current required so save energy. Neat find anyways though.

Attached is the file to playa round... note I put an initial charge on C1 for the sake of run time.
Here's where I was probing it as far as voltages (the current measurement is through the coil):